Multiple linear regression efforts to model the relationship between two or more explanatory variables and a response variable by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y. The population regression line for p explanatory variables x1, x2… xp is defined to be μy=β°+β1x1+β2x2+ … + βpxp

This line describes how the mean response f changes with the explanatory variables. The observed values for y vary about their means μyand are assumed to have the same standard deviation σ. The fitted values b0, b1... bp estimate the parameters β1,β2,…,βp of the population regression line. Formally, the model for multiple linear regressions, given n observations, is yi= β0 + β1xip + β2xip + ... βpxip + εi for i= 1,2, ... n. In the least-squares model, the best-fitting line for the observed data is calculated by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values. The least-squares estimates b0, b1, ... bp are usually computed by statistical software. The values fit by the equation b0 + b1xi1 + ... + bpxip are denoted yl, and the residuals ei are equal to yi - yi, the difference between the observed and fitted values. The sum of the residuals is equal to zero. The variance σ2 may be estimated by s2= ei2n-p-1 , also known as the mean-squared error (or MSE). The estimate of the standard error s is the square root of the MSE. Nowadays, there are many computer languages that can be used for computer programming. Some of them are C and C++. C++ is a type of computer programming language. Created in 1983 by Bjarne Stroustrup, C++ was designed to serve as an enhanced version of the C programming language. C++ is object oriented and is considered a high level language. However, it features low level facilities. C++ is one of the most commonly used programming languages. The development of C++ actually began four years before its release, in 1979. It did not start out with the name C++; its first name was C with Classes. In the late part of 1983, C with Classes was first used for AT&T’s internal programming needs. Its name was changed to C++ later in the same year. C++ was not released commercially until the late part of 1985. Developed at Bell Labs, C++ enhanced the C programming language in a variety of ways. Among the features of C++ are classes, virtual functions, templates, and operator overloading. The C++ language also counts multiple inheritance and exception handling among its many features. C++ introduced the use of declarations as statements and includes more type checking than is available with the C programming language. Considered a superset of the C programming language, C++ maintains a variety of features that are included within its predecessor. As such, C programs are generally able to run successfully in C++ compilers. However, there are some issues that may cause C code to perform differently in C++ compilers. In fact, it is possible for some C code to be incompatible in C++. Library creation is cleaner in C++. The C++ programming language is considered portable and does not require the use of a specific piece of hardware or just one operating system. Another important feature of C++ is the use of classes. Classes help programmers with the organization of their code. They can also be beneficial in helping programmers to avoid mistakes. However, there are times when mistakes do slip through. When this happens, classes can be instrumental in finding bugs and correcting them. The original C++ compiler, called Cfront, was written in the C++ programming language. C++ compilation is considered efficient and fast. Its speed can be...

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...Linear -------------------------------------------------
Important
EXERCISE 27 SIMPLE LINEARREGRESSION
STATISTICAL TECHNIQUE IN REVIEW
Linearregression provides a means to estimate or predict the value of a dependent variable based on the value of one or more independent variables. The regression equation is a mathematical expression of a causal proposition emerging from a theoretical framework. The linkage between the theoretical statement and the equation is made prior to data collection and analysis. Linearregression is a statistical method of estimating the expected value of one variable, y, given the value of another variable, x. The term simple linearregression refers to the use of one independent variable, x, to predict one dependent variable, y.
The regression line is usually plotted on a graph, with the horizontal axis representing x (the independent or predictor variable) and the vertical axis representing the y (the dependent or predicted variable) (see Figure 27-1). The value represented by the letter a is referred to as the y intercept or the point where the regression line crosses or intercepts the y-axis. At this point on the regression line, x = 0. The value represented by the letter b is referred to as the slope, or the coefficient of x. The slope determines the...

...Linear-Regression Analysis
Introduction
Whitner Autoplex located in Raytown, Missouri, is one of the AutoUSA dealerships. Whitner Autoplex includes Pontiac, GMC, and Buick franchises as well as a BMW store. Using data found on the AutoUSA website, Team D will use LinearRegression Analysis to determine whether the purchase price of a vehicle purchased from Whitner Autoplex increases as the age of the consumer purchasing the vehicle increases. The data set provided information about the purchasing price of 80 domestic and imported automobiles at Whitner Autoplex as well as the age of the consumers purchasing the vehicles. Team D selected the first 30 of the sampled domestic vehicles to use for this test. The business research question Team D will answer is: Does the purchase price of a consumer increase as the age of the consumer increases? Team D will use a linear-regression analysis to test the age of the consumers and the prices of the vehicles.
Five Step Hypothesis Testing
Team D will conduct the two-sample hypothesis using the following five steps:
1. Formulate the hypothesis
2. State the decision rule
3. Calculate the Test Statistic
4. Make the decision
5. Interpret the results
Step 1- Formulate the Hypothesis
Using the research question: Does the purchase price of an automobile purchased at Whitner Autoplex, increase as the age of the...

...Topic 8: MultipleRegression Answer
a.
Scatterplot
120 Game Attendance 100 80 60 40 20 0 0 5,000 10,000 15,000 20,000 25,000 Team Win/Loss %
There appears to be a positive linear relationship between team win/loss percentage and
game attendance. There appears to be a positive linear relationship between opponent win/loss percentage and game attendance.
There appears to be a positive linear relationship between games played and game
attendance. There does not appear to be any relationship between temperature and game attendance.
b. Game Attendance Game Attendance Team Win/Loss % Opponent Win/Loss % Games Played Temperature Team Win/Loss % Opponent Win/Loss % Games Played Temperature
1 0.848748849 1 0.414250332 0.286749997 1 0.599214835 0.577958172 0.403593506 1 -0.476186226 -0.330096097 -0.446949168 -0.550083219
1
No alpha level was specified. Students will select their own. We have selected .05. Critical t = + 2.1448 t for game attendance and team win/loss % = 0.8487/ (1 − 0.84872) /(16 − 2) = 6.0043 t for game attendance and opponent win/loss % = 0.4143/ (1 − 0.41432) /(16 − 2) = 1.7032 t for game attendance and games played = 0.5992/ (1 − 0.59922) /(16 − 2) = 2.8004 t for game attendance and temperature = -0.4762/ (1 − ( − 0.4762 ) ) /(16 − 2) = -2.0263 There is a significant relationship between game attendance and team win/loss % and games played. Therefore a...

...Multipleregression: OLS method
(Mostly from Maddala)
The Ordinary Least Squares method of estimation can easily be extended to models involving two or more explanatory variables, though the algebra becomes progressively more complex. In fact, when dealing with the general regression problem with a large number of variables, we use matrix algebra, but that is beyond the scope of this course.
We illustrate the case of two explanatory variables, X1 and X2, with Y the dependant variable. We therefore have a model
Yi = α + 1X1i + 2X2i + ui
Where ui~N(0,σ2).
We look for estimators so as to minimise the sum of squared errors,
S =
Differentiating, and setting the partial differentials to zero we get
=0 (1)
=0 (2)
=0 (3)
These three equations are called the “normal equations”. They can be simplified as follows: Equation (1) can be written as
or
(4)
Where the bar over Y, X1 and X2 indicates sample mean. Equation (3) can be written as
Substituting in the value of from (4), we get
(5)
A similar equation results from (3) and (4). We can simplify this equation using the following notation. Let us define:
Equation (5) can then be written
S1Y = (6)
Similarly, equation (3) becomes
S2Y = (7)
We can solve these two equations to get:
and
Where =S11S22 – S122. We may therefore obtain from equation (4).
We can...

...Topic 4. Multipleregression
Aims
• Explain the meaning of partial regression coefficient and calculate and interpret multipleregression models • Derive and interpret the multiple coefficient of determination R2and explain its relationship with the the adjusted R2 • Apply interval estimation and tests of significance to individual partial regression coefficients d d l ff • Test the significance of the whole model (F-test)
Introduction
• The basic multipleregression model is a simple extension of the bivariate equation. • By adding extra independent variables, we are creating a multiple-dimensioned space, where the model fit is a some appropriate space. , p , • For instance, if there are two independent variables, we are fitting the points to a ‘plane in space’. trick. • Visualizing this in more dimensions is a good trick
Model specification – scalar version
• The basic linear model: • Yi = ß0 + ß1 X1i+ ß2X2i+ ß3X3i +….+ ßkXki +ui …. u • If bivariate regression can be described as a line on a plane, multipleregression represents a k-dimensional object in a k+1 d dimensional space. l
Matrix version
• We can use a different type of mathematical g structure to describe the regression model Frequently called Matrix or Linear Algebra •...

...both sides of the linearregression line. When the incomes of the consumer increase the sales for cars also rises presenting a positive result. Therefore, as long as the incomes continue to grow the relationship to car sales will also trend to the right in an upward, positive motion.
B. What is the direction of causality in this relationship - i.e. does having a more expensive car make you earn more money, or does earning more money make you spend more on your car? In other words, define one of these variables as your dependent variable (Y) and one as your independent variable (X).
Depending on the each individuals perspective the variable can switch between dependent and independent based on the person’s viewpoint. For this purpose, the independent variable which is represented by (X) is the annual income. The dependent variable is represented by (Y) and is the cost of the car. The reason I chose to have the annual income as the independent variable is because a person will continue to look for a job with security, growth potential, and a higher income. The car is seen as a vehicle of transportation only and needed to get to work and home. It is a necessity, but not a luxury item with elaborate expenses. We can have the basic model without all the bells and whistles to accomplish the task to get to and from a location.
C. What method do you think would be best for testing the relationship between your dependent and independent variable,...

...the hedonic regression. This method is specific to breaking down items that are not homogenous commodities, to estimate value of its characteristics and ultimately determine a price based on the consumers’ willingness to pay. The approach in estimating the values is done by measuring the differences in the price of certain goods with regards to specific location. E.g. average cost of real estate is much lower in Missouri than in California. Location may be the biggest factor in real estate pricing.
2. Data and Regression Analysis
My data is for Blowing Rock, NC. It’s a resort town in the Blue Ridge Mountains. The attractions here are mostly outdoor activities taking place in the secluded wilderness. The population is only about 1500 and the average cost of a house from my data is $485,839.50.
For my linearregression, I am modeling the relationship between the price of homes, being my dependent variable, and some characteristics of the homes, being my explanatory variables. Originally my data consisted of the following for real estate in Blowing Rock, NC: price - selling price, miles from central business district, number of bedrooms, number of full bathrooms, number of half bathrooms, the year the home was built, square footage, number of garages, whether or not the house was located in a subdivision, lot size, if the house had a good view, number of days on the market, and difference between asking price...